Algebra Examples

f (x) = x (x - 2) (x + 2)

$f (x) = x (x - 2) (x + 2)$

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Simplify by multiplying through.

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Step 1.1.1.1

Apply the distributive property.

(x \cdot x + x \cdot - 2) (x + 2)

$(x \cdot x + x \cdot - 2) (x + 2)$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Multiply

x

$x$ by

x

$x$ .

(x^{2} + x \cdot - 2) (x + 2)

$(x^{2} + x \cdot - 2) (x + 2)$

Step 1.1.1.2.2

Move

- 2

$- 2$ to the left of

x

$x$ .

(x^{2} - 2 \cdot x) (x + 2)

$(x^{2} - 2 \cdot x) (x + 2)$

(x^{2} - 2 \cdot x) (x + 2)

$(x^{2} - 2 \cdot x) (x + 2)$

(x^{2} - 2 \cdot x) (x + 2)

$(x^{2} - 2 \cdot x) (x + 2)$

Step 1.1.2

Expand

(x^{2} - 2 x) (x + 2)

$(x^{2} - 2 x) (x + 2)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

x^{2} (x + 2) - 2 x (x + 2)

$x^{2} (x + 2) - 2 x (x + 2)$

Step 1.1.2.2

Apply the distributive property.

x^{2} x + x^{2} \cdot 2 - 2 x (x + 2)

$x^{2} x + x^{2} \cdot 2 - 2 x (x + 2)$

Step 1.1.2.3

Apply the distributive property.

x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ by adding the exponents.

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Step 1.1.3.1.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

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Step 1.1.3.1.1.1.1

Raise

x

$x$ to the power of

1

$1$ .

x^{2} x^{1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2} x^{1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 1.1.3.1.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 1.1.3.1.1.2

Add

2

$2$ and

1

$1$ .

x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 1.1.3.1.2

Move

2

$2$ to the left of

x^{2}

$x^{2}$ .

x^{3} + 2 \cdot x^{2} - 2 x \cdot x - 2 x \cdot 2

$x^{3} + 2 \cdot x^{2} - 2 x \cdot x - 2 x \cdot 2$

Step 1.1.3.1.3

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 1.1.3.1.3.1

Move

x

$x$ .

x^{3} + 2 x^{2} - 2 (x \cdot x) - 2 x \cdot 2

$x^{3} + 2 x^{2} - 2 (x \cdot x) - 2 x \cdot 2$

Step 1.1.3.1.3.2

Multiply

x

$x$ by

x

$x$ .

x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2

$x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2$

x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2

$x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2$

Step 1.1.3.1.4

Multiply

2

$2$ by

- 2

$- 2$ .

x^{3} + 2 x^{2} - 2 x^{2} - 4 x

$x^{3} + 2 x^{2} - 2 x^{2} - 4 x$

x^{3} + 2 x^{2} - 2 x^{2} - 4 x

$x^{3} + 2 x^{2} - 2 x^{2} - 4 x$

Step 1.1.3.2

Subtract

2 x^{2}

$2 x^{2}$ from

2 x^{2}

$2 x^{2}$ .

x^{3} + 0 - 4 x

$x^{3} + 0 - 4 x$

Step 1.1.3.3

Add

x^{3}

$x^{3}$ and

0

$0$ .

x^{3} - 4 x

$x^{3} - 4 x$

x^{3} - 4 x

$x^{3} - 4 x$

x^{3} - 4 x

$x^{3} - 4 x$

Step 1.2

The largest exponent is the degree of the polynomial.

3

$3$

3

$3$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Simplify by multiplying through.

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Step 3.1.1.1

Apply the distributive property.

(x \cdot x + x \cdot - 2) (x + 2)

$(x \cdot x + x \cdot - 2) (x + 2)$

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Multiply

x

$x$ by

x

$x$ .

(x^{2} + x \cdot - 2) (x + 2)

$(x^{2} + x \cdot - 2) (x + 2)$

Step 3.1.1.2.2

Move

- 2

$- 2$ to the left of

x

$x$ .

(x^{2} - 2 \cdot x) (x + 2)

$(x^{2} - 2 \cdot x) (x + 2)$

(x^{2} - 2 \cdot x) (x + 2)

$(x^{2} - 2 \cdot x) (x + 2)$

(x^{2} - 2 \cdot x) (x + 2)

$(x^{2} - 2 \cdot x) (x + 2)$

Step 3.1.2

Expand

(x^{2} - 2 x) (x + 2)

$(x^{2} - 2 x) (x + 2)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

x^{2} (x + 2) - 2 x (x + 2)

$x^{2} (x + 2) - 2 x (x + 2)$

Step 3.1.2.2

Apply the distributive property.

x^{2} x + x^{2} \cdot 2 - 2 x (x + 2)

$x^{2} x + x^{2} \cdot 2 - 2 x (x + 2)$

Step 3.1.2.3

Apply the distributive property.

x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2} x + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ by adding the exponents.

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Step 3.1.3.1.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

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Step 3.1.3.1.1.1.1

Raise

x

$x$ to the power of

1

$1$ .

x^{2} x^{1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2} x^{1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 3.1.3.1.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{2 + 1} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 3.1.3.1.1.2

Add

2

$2$ and

1

$1$ .

x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2

$x^{3} + x^{2} \cdot 2 - 2 x \cdot x - 2 x \cdot 2$

Step 3.1.3.1.2

Move

2

$2$ to the left of

x^{2}

$x^{2}$ .

x^{3} + 2 \cdot x^{2} - 2 x \cdot x - 2 x \cdot 2

$x^{3} + 2 \cdot x^{2} - 2 x \cdot x - 2 x \cdot 2$

Step 3.1.3.1.3

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 3.1.3.1.3.1

Move

x

$x$ .

x^{3} + 2 x^{2} - 2 (x \cdot x) - 2 x \cdot 2

$x^{3} + 2 x^{2} - 2 (x \cdot x) - 2 x \cdot 2$

Step 3.1.3.1.3.2

Multiply

x

$x$ by

x

$x$ .

x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2

$x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2$

x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2

$x^{3} + 2 x^{2} - 2 x^{2} - 2 x \cdot 2$

Step 3.1.3.1.4

Multiply

2

$2$ by

- 2

$- 2$ .

x^{3} + 2 x^{2} - 2 x^{2} - 4 x

$x^{3} + 2 x^{2} - 2 x^{2} - 4 x$

x^{3} + 2 x^{2} - 2 x^{2} - 4 x

$x^{3} + 2 x^{2} - 2 x^{2} - 4 x$

Step 3.1.3.2

Subtract

2 x^{2}

$2 x^{2}$ from

2 x^{2}

$2 x^{2}$ .

x^{3} + 0 - 4 x

$x^{3} + 0 - 4 x$

Step 3.1.3.3

Add

x^{3}

$x^{3}$ and

0

$0$ .

x^{3} - 4 x

$x^{3} - 4 x$

x^{3} - 4 x

$x^{3} - 4 x$

x^{3} - 4 x

$x^{3} - 4 x$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

x^{3}

$x^{3}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

1

$1$

1

$1$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and rises to the right

Step 7